• Cracked the engima machine in WWII

• Nate Silver, statistician and found of 538.org, successfully predicted the outcome of the 2 Obama elections, and the
  • Using polls, demographic data, other metrics.

What do Alan Turing and Nate Silver have in common?

• Worked on complex problems
• Large inherent uncertainty that needed to be quantified
• Requires efficient integration of many sources of information
• Used Bayesian Analysis.

Method for figuring out unknowns that requires 3 things

1. Data
2. A generative model: any model we can simulate data from
3. Priors: existing information or current knowledge

When we know the parameters we can simulate data

Using Bayesian inference, we can estimate parameters from data

To understand it we need to know about

• Probability
• Bayes’ theorem
• Likelihood, Posteriors, and Priors

• Probability is the chance of an event occurring, defined between 0 and 1.

  • 0 = will never happen, 1 = will always happen

• Random Variable is the variable that takes values depending on outcomes of random events.

  • Discrete (e.g. dead/alive) or Continuous (e.g. age, weight).

• Probability distribution describes how the probabilities are distributed over the values of the random variable.

• To define a probability need

  • To identify all possible outcomes (events), i.e. sample space
  • Calculate the frequency the event occurs

• \(P()\), probability.
• \(P(A)\), probability of Event A.
• \(P(B)\), probability of Event B.
• \(P(A \cup B)\) The probability of Event A or the probability of Event B occurring
• \(P(A \cap B)\) The probability of both Event A and Event B occurring.
• \(P(A | B)\) The probability of Event A given Event B has already occurred

• The probability must be non-negative for each value of the random variable and the sum of probabilities must equal 1.

• If two events (A and B) are mutually exclusive, the probability that either occurs is the sum of their individual probabilities \(P(A \cup B)\):

  • e.g. \(P(\text{HIV} \cup \text{HIV-})\) = \(P(\text{HIV}) + P(\text{HIV-})\).

• If two events are not mutually exclusive, we need to correct for double counting:

  • e.g. \(P(\text{HIV or Female})\) = \(P(\text{HIV})\) + \(P(\text{Female})\) - \(P(\text{HIV} \cap \text{Female})\)

• The conditional probability of \(A\) given \(B\), \(P(A|B)\),
  • \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
• Conditional probability of an individual having HIV and being Female is:
  • \(P(\text{HIV | Female}) = \frac{P(\text{HIV} \cap \text{ Female})}{P(\text{Female})}\)

• Bayes’ Rule is a recipe for turning around a conditional probability so we can learn about the thing we are interested in.

• Use Bayes’ Rule to go from the likelihood \(P(\text{Data}|\text{Hypothesis})\) to the information we want:

  • \(P(\text{Hypothesis}|\text{Data})\).

• Bayes’ Theorem:

  • \(P(\text{Hypothesis}|\text{Data}) = \frac{P(\text{Data}|\text{Hypothesis})P(\text{Hypothesis})}{P(\text{Data})}\)

• Probability of having the Zombie virus \(I\) is \({\color{red}{P(\text{Infected})}} = 10^{-6}\)
• Zombie Test has no false negatives: \({\color{blue}{P(\text{Test positive}|\text{Infected})}} = 1\).
• Some false positives: \({\color{orange}{P(\text{Test positive}| \text{Uninfected})}} = 10^{-2}\)

• Bayes’ Theorem:
  • \(P(\text{Hypothesis}|\text{Data}) = \frac{P(\text{Data}|\text{Hypothesis})P(\text{Hypothesis})}{P(\text{Data})}\)
• Replacing Hypothesis with “Infected” and Data with “Test positive”:
  • \(P(\text{Infected}|\text{Test positive}) = \frac{P(\text{Test positive|Infected})P(\text{Infected})}{P(\text{Test positive})}\)
• Unknown probability of Testing Positive \(P(\text{Test positive})\) in the population?

Probability of Testing Positive: \(P(\text{Test Positive})\)

• 2 groups will test positive: Infected and Uninfected

• \(= P(\text{Test Positive} \cap \text{Infected}) + P(\text{Test Positive} \cap \text{Uninfected})\) which is equivalent to

• \(= P(\text{Test positive}|\text{Infected})P(\text{Infected})\) + \(P(\text{Test positive|Uninfected})P(\text{Uninfected})\)

• Being infected and uninfected are mutually exclusive so

• \(P(\text{Uninfected}) = 1-P(\text{Infected})\)

• Probability of being infected: \({\color{red}{P(\text{Infected})}} = 10^{-6}\)
• Probability of testing positive given you are infected: \({\color{blue}{P(\text{Test positive}|\text{Infected})}} = 1\).
• Probability of testing postiive given you are uninfected \({\color{orange}{P(\text{Test positive}| \text{Uninfected})}} = 10^{-2}\)

Probability of Testing Positive: swapping in what we know

• \(= {\color{blue}{P(\text{Test positive}|\text{Infected})}}{\color{red}{P(\text{Infected})}}\) + \({\color{orange}{P(\text{Test positive}|\text{Uninfected})}}(1 - {\color{red}{P(\text{Infected})}})\)

• Gives us: \(= {\color{blue}{1}}\times{\color{red}{10^{-6}}} + {\color{orange}{10^{-2}}}\times(1 - {\color{red}{10^{-6}}})\)

• \(=10^{-6} + 10^{-2}+ 10^{-8}\)

• \({\color{green}{\approx 10^{-2}}}\)

\(P(\text{Infected}|\text{Test positive}) = \frac{{\color{blue}{P(\text{Test positive|Infected})}}{\color{red}{P(\text{Infected}}})}{{\color{green}{P(\text{Test positive})}}}\)

• If we plug in what we know:

  • \({\color{blue}{P(\text{Test positive|Infected})}}\) = 1
  • \({\color{red}{P(\text{Infected})}}\) = \(10^{-6}\)
  • \({\color{green}{P(\text{Test positive})}}\) = \(10^{-2}\)

• \(\frac{{\color{blue}1}\times{\color{red}{10^{-6}}}}{{\color{green}{10^{-2}}}} = 0.00001\)

• Phew! Even though false positives are unlikely, the chance you are a zombie is roughly 1 in 10,000.

Bayes’ Rule is general, our hypothesis and data can be any events. We are often using it in our models to assess our parameters.

\[ P(Parameters \mid Data) = \frac{P(Data \mid Parameters) \, P(Parameters)}{P(Data)} \] - Prior Knowledge + Current Data = Current Knowledge

• Prior distribution: what you know about the parameters, before getting the data - \(P(Parameters)\).

• Likelihood: based on modeling assumptions, how likely are the data if the chosen values for our parameters are correct - \(P(Data|Parameters)\)

• Posterior: probability of the chosen parameters given the Data, \(P(Parameters|Data)\).

• We can keep adding data and updating the posterior (our current knowledge of the process) as data becomes available.

• Priors come from all information external to the current study (i.e. everything else)
  • Other studies, expert opinion, experiments etc.
  • We may inherently know something about the prior
    • e.g. it can’t be less than 0, or that it comes from a gamma distribution.
  • Flat, “uninformative” priors

Draw a prior for the percentage of the earth that is water.

Who would win in a fight between Thomas Bayes and Ronald Fisher?

• Frequentist statistics
  • Classical statistics, p-values, null hypotheses, repeated experiments.
    • e.g. probability explained flipping coins
  • Assumes that there is a “true” state of the world which gives rise to a distribution of possible experimental outcomes.
  • In frequentist view of probability, probability depends on a series of hypothetical repeated experiments.
  • The underlying parameters and probabilities remain constant during this repeatable process.

• Bayesian Statistics
  • Assumes the data is truth, while the parameter values or hypotheses have probability distributions.
  • In a Bayesian view of probability, we interpret the probability distribution as quantifying uncertainty about the world.
  • In Bayesian statistics, answers depend on what was observed (e.g. the data).
  • Allow us to make statements about the probability of different hypotheses or parameter values.

1. Can incorporate prior observations and prior knowledge

2. Useful in complex models with missing data and several layers of variability

3. Can be used to make management decisions based on our data, including evaluating unlikely scenarios.

• Computationally expensive.
• In Bayesian statistics, we have to specify our prior beliefs about the probability of different hypotheses.
  • These prior beliefs actually affect our answers.
  • You can get back what you put in and confirm your hypotheses.
• To deal with subjectivity
  • Set a “flat prior” or uninformative prior to let the data speak for themselves.
  • Sensitivity analyses: try different priors and see whether it influences the results.
  • Use of “penalized-complexity priors”

• Ben Bolker, Ecological Models and Data in R
• Introduction to Bayesian Data Analysis by Rasmus Bååth https://www.youtube.com/watch?v=3OJEae7Qb_o